where \(\mathcal {E}_\text{cr}\)  is the magnetic energy at the beginning of the nonlinear stage at the time \(t = t_\text{cr}\) .
In a system with the initial magnetic energy \(\mathcal {E}_\text{cr}\)  in equipartition with the turbulent energy at some length
scale within the inertial range of turbulence, that is, the turbulence is super-Alfvénic (see Section REF ),
we have \(t_\text{cr} = 0\) .
In a system with an initially very weak magnetic field, the turbulent dynamo is first in the kinematic regime until the energy
equipartition between turbulence and magnetic field can be reached at some length scale within the inertial range.
In this case, \(t_\text{cr}\)  is given by the duration of the kinematic stage and depends on the plasma conditions
(XL16).
The linear dependence of \(\mathcal {E}\)  on time comes from the constant energy transfer rate \(\epsilon = v_k^3 k\)  of turbulence.
Here \(v_k\)  is the turbulent velocity corresponding to the wavenumber \(k\) .
The turbulent energy cascade with the turbulent kinetic energy
transferred from
one eddy to the next smaller eddy has a constant energy transfer rate.
Meanwhile, a small fraction \(3/38\)  of the turbulent energy is converted to magnetic energy within the eddy turnover time,
resulting in a constant magnetic energy growth rate.
The low efficiency of nonlinear dynamo growth is the consequence of reconnection diffusion.
The growing magnetic energy is passed down toward smaller scales along the turbulent energy cascade.
The above small growth rate of nonlinear dynamo is quantitatively consistent with that measured in nonlinear dynamo simulations by, e.g.,
[1]}, [2]}.
